Scattering of X-rays
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Transcript of Scattering of X-rays
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Scattering of X-rays
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Books on SAS
- " The origins" (no recent edition) : Small Angle Scattering of X-raysA. Guinier and A. Fournet, (1955), in English, ed. Wiley, NY
- Small-Angle X-ray ScatteringO. Glatter and O. Kratky (1982), Academic Press.
- Structure Analysis by Small Angle X-ray and Neutron ScatteringL.A. Feigin and D.I. Svergun (1987), Plenum Press.
- Neutron, X-ray and Light scattering: Introduction to an investigative Tool for Colloidal and Polymeric Systems (1991),
P. Lindner and Th. Zemb (ed.), North-Holland
-The Proceedings of the SAS Conferences held every three years are usually published in the Journal of Applied Crystallography. -The latest proceedings are in the J. Appl. Cryst., 30, (1997), next one soon to come out
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Reminder - notations
Fourier Transform
2F( ) ( ) i
Ve dV
r
rsrs r(r)
F. T.
2( ) F( ) i
Ve dV
S
rsSr sF(s)
F. T.-1
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Properties of the Fourier Transform
- 1 – linearity
FT (+ ) = FT(FT()
-2 – value at the origin
F(0) ( )V
dVr
rr
- 3 – translation
2F( ) ie Rss(r + R) F. T.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Fourier Transforms of simple functions
- 1 – delta function at the origin
(r)=(r)
-2 – Gaussian of width 1/K2
2F( )
sKs e
- 3 – rectangular function
2 2
( ) K rf r e
- 1 – constant unitary value
F(s)=1
-2 – Gaussian of width K
- 3 – sinc
sinF( )
sas a
sa
1/a-1/a
a
1
-a/2 a/2
f(x)
1)
o rs
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
r r
B(r)
A(r) A(r)*B(r)
rArB
1
Convolution product
A( ) B( ) A( )B( )V
dV u
ur r u r u
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Convolution product
u u
B(r-u)
A(u) A(r)*B(r)
rArB rA + rB
rA - rB
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Fundamental Propertyof the convolution product
FT(A B) FT(A) FT(B)
FT(A B) FT(A) FT(B)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Autocorrelation function
( ) ( ) ( ) ( ) ( )V
dV u
ur r r r u u
r
particle ghost
0 ( ) ( ) (0)r r characteristic function
r
(r)
spherical average ( ) ( )r r
(r)= (uniform density) => (0)= V
1
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Distance distribution function
2 2 20( ) ( ) ( )p r r r r r V
rij ji
r
p(r)
Dmax
(r) : probability of finding a point at r from a given point
number of el. vol. i V - number of el. vol. j 4r2
number of pairs (i,j) separated by the
distance r 4r2V(r)=(4/2)p(r)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
The elastically scattered intensity is given by the Thomson formula2
20 0 2
1 1 cos 2
2I r I
d
2 : scattering angle, cos2 close to 1 at small-angles
I0 intensity (energy/unit area /s) of the incident beam.The scattered beam has an intensity Ie/d2 at a distance d from the scattering electron. In what follows, is omitted and only the number of electronsis considered
2 26 20 7.95 10 cmeI r is the scattering cross-section of the electron
20r
Scattering by a free electron
X-ray scattering length of an electron 2
130 2
2.82 10 cme
rmc
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Scattering by an electron at a position r
source
s0
s0
s1
s1
detector
r
r.s0
r.s1
Path difference = r.s1-r.s0 = r.(s1 - s0) or r.(s1 - s0)/ in cycles, for X-rays of wavelength
2
O
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
the scattering vector s
0 1
1
s s
2sins
s
s1
s0
O
s
length 1/
length 1/
2
scattered
1 0 s s ss is called the scattering vector
The scattered amplitude by the electron at r iswhere E(s) is the scattered amplitude by an electron at the origin
2 .( ) iE s e r s
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
scattering vector
2sin 2s
s
SAXS measurements are restricted to small-angles :2 5 deg = 85 mrad at = 1.5 ÅIn this case, the further approximation is valid :
4 sin" "q h s
!
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Scattered amplitude
2F( ) ( ) i
Ve dV
r
rsrs r
2sin
s
Scattered intensity
I( ) F( ).F ( )s s s
Remark
0s F(0) ( )V
dVr
rr
For a particle I(0) = m2 m : number of electrons
F(s) is the Fourier transform from the electron density (r) describing the scattering object
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Solution X-ray scattering
2X-ray beam
Sample
Detector
Diagram of the experimental set-up
s = 2 sin/
X-ray scattering curve
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Particles in solution
The relevant quantity is now the
contrast of electron density
between the particle and the solvent
(r) = (r) - 0
with 0 = 0.3346 el. A-3 the electron density of water
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Contrast of electron density
solvent
particle 0.43
0 0.335
el. A-3
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
X-ray scattering power of a protein solution
Let W be the probability of an incident photon being scattered by a solution of spherical proteins :
using the expression for the optimal thickness d(cm) = 0.3/2 (A3)
Protein : solvent
Example : 10 mg/ml solution of myoglobin v= 0.74 cm3.g-1 r=20A, = 1.5A
W=2 10-5 (x efficiency x geometrical factor)
30.43 .el A 30 0.335 .el A
3 10 20.1 . 2.8 10el A cm
2 2( ) 0.3( ) /W cv dr cvr
42.3 10 /W cvr
from H.B. StuhrmannSynchrotron Radiation ResearchH. Winick, S. Doniach Eds. (1980)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Solution of particles
=
=Solution
(r)
F(c,s)
Motif (protein)
p(r)
F(0,s)
Lattice
d(r)
(c,s)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Solution of particles
For spherically symmetrical particles
I(c,s) = I(0,s) x S(c,s)
form factorof the particle
structure factorof the solution
Still valid for globular particles though over a restricted s-range
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Solution of particles
- 1 – monodispersity: identical particles
- 2 – size and shape polydispersity
- 3 – ideality : no intermolecular interactions
- 4 – non ideality : existence of interactionsbetween particles
In the following, we make the double assumption 1 and 3
2 and 4 dealt with at a later stage in the course
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Ideal and monodisperse solution
Particles in solution => thermal motion => particles have a random orientation / X-ray beam. The sample is isotropic. Therefore, only the spherical average of the scattered intensity is experimentally accessible.
21F ( ) ( ) i
Ve dV
r
rsrs r
1 1 1 1( ) ( ) F ( ).F ( )i s i s s s
Ideality and monodispersity 1I( ) i ( )s sN
I( ) I( ) F( ).F ( )s s s s
time
particles
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
1( ) [ ( )]. [ ( )] [ ( )* ( )]i s FT FT FT r r r r
21( ) [ ( )] ( ) i
Vi s FT e dV
r
rsrr r
Let us use the properties of the Fourier transform and of the convolution product
1 0
sin(2 )( ) 4 ( )
2
rsi s p r dr
rs
2( ) ( )p r r rwith
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Or one of the symmetrical expressions :
1 0( ) 2 ( )sin(2 )si s r r rs dr
0
1( ) I( )sin(2 )r r s s rs ds
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Relationships real space – reciprocal space
(r)
F(s) I(s) I(s)
p(r) p(r)
FTFT FT
*
.
< >
< >
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
If the particle is described as a discrete sum of elementaryscatterers,(e.g. atoms) the scattered intensity is :
Debye formula
where the fi(s) are the atomic scattering factors, and the spherically averaged intensity is (Debye) :
N N2
11 1
( ) ( ) ( ) i ji
i ji j
i f f e
r r ss s s
N N
11 1
sin 2( ) ( ) ( )
2ij
i ji j ij
r si s f s f s
r s
where ij i jr r r
The Debye formula is widely used for model calculations
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
1 2
121 1 2 1 2
12
sin 2( ) ( ) ( )
2V V
r si s d d
r s
r r
r r r r
Debye formula
If the particle is described using a density distribution, the Debye formula is written :
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Porod invariant
2
0
sin(2 )( ) 2 I( )
2
rsr s s ds
rs
2
0(0) 2 I( ) 2s s ds Q
Q is called the Porod invariant
Q depends on the mean square electron density contrast
For r=0 :
2 2
0(0) ( ) dV V
rrBy definition :
22
0( )
2
VQ s I s ds
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Intensity at the origin
'1(0) ( ) ( )
V Vi dV dV
r rr r'r r'
2
2 21 0 0(0) ( ) ( )P
A
Mi m m m v
N
A
NMc
N V is the concentration (w/v), e.g. in mg.ml-1
2
0(0) ( )PA
cMVI v
N
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Intensity at the origin
If : the concentration c (w/v),the partial specific volume ,the intensity on an absolute scale, i.e. the number of incident photons
are known,
Then, the molecular weight of the particle can be determinedfrom the value of the intensity at the origin
Pv
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Intensity on an absolute scale
The experimental intensity Iexp(0) is expressed as :
20
exp 02(0) ( )e
PA
I I cMdI v
N a
d : thickness of the samplea : sample-detector distance
scattering cross-section of the electronI0 total energy of the incident beam
2 26 20 7.95 10eI r cm
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Intensity on an absolute scale
The determination of I0 is performed using: - attenuation of the direct beam by calibrated filters(valid in the case of monochromatic radiation, beware of harmonics)
- by using a well-known reference sample like the Lupolen (Graz)- using the scattering by water as proposed by O. Glatter inD. Orthaber, A. Bergmann & O. Glatter, J. Appl. Cryst. (2000), 33, 218-225 Thorough presentation of calculations on an absolute scale
P. Bösecke & O. Diat J. Appl. Cryst. (1997), 30, 867-871 Clear presentation of the geometrical calculations and all the procedures used on ID2 (ESRF) to put intensities on an absolute scale.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Guinier law
For small values of x, sinx/x can be expressed as :
idealmonodisperse
2 4sin 2 2 2
1 ...2 3! 5!
rs rs rs
rs
0
sin(2 )( ) 4 ( )
2
rsI s p r dr
rs
Hence, close to the origin:
22with and
4 (0) 4 ( ) = ( ) ( )
6I p r dr k r p r dr p r dr
22( ) (0) 1 (0) ksI s I ks I e
The scattering curve of a particle can be approximated by a Gaussian curve in the vicinity of the origin
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
1 2
121 2 1 2
12
sin 2( ) ( ) ( )
2V V
r sI s d d
r s
r r
r r r r
Guinier law
Let us introduce the expansion of sinx/x into the Debye formula :
We obtain the following expression :
1 2 1 2
2 22
1 2 1 2 1 2 1 2 1 2
4( ) ( ) ( ) ( ) ( )
6V V V V
sI s d d d d
r r r r
r r r r r r r r r r
1 2
1 2
22 1 2 1 2 1 2
1 2 1 2
hence ( ) ( )4
6
)
( ) (
V V
V V
d dk
d d
r r
r r
r r r r r r
r r r r
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Guinier law
Radius of gyration
idealmonodisperse
22 2
2it comes 2 ( )4 4
6 3( )
Vg
V
r dVk R
dV
r
r
r
r
r
r
1 2 1 0 0 2writing r r r r r r where r0 is the center of mass
Guinier law24
3
2 2gI(s) I(0)exp R s
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Radius of gyration
Radius of gyration :
2
2( )
( )V
g
V
r dVR
dV
r
r
r
r
r
r
Rg is the quadratic mean of distances to the center of mass weighted by the contrast of electron density.Rg is an index of non sphericity.For a given volume the smallest Rg is that of a sphere :
Ellipsoïd of revolution (a, b) Cylinder (D, H)
3
5gR R
2 22
5g
a bR
2 2
8 12g
D HR
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Guinier plotexample
Validity range : 0 < 2Rgs<1.2For a sphere
The law is generally used under its log form :
24
3
2 2
gln I(s) ln I(0) R s
A linear regression yields two parameters : I(0) (y-intercept)
Rg from the slope
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Attractive Interactions
-crystallins c=160 mg/ml in 50mM Phosphate pH 7.0
I(s
)
0
1 104
2 104
3 104
4 104
5 104
0 0.01 0.02
30C
25C
20C
15C
10C
s = 2(sinA-1A. Tardieu et al., LMCP (Paris)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Repulsive Interactions
ATCase in 10mM borate buffer pH 8.3
0
500
1000
1500
2000
2500
3000
3500
4000
0 0.005 0.01 0.015 0.02
92,5mg/ml31,5 mg/ml13,3 mg/ml6,6 mg/ml3,3 mg/ml
I(s)
s = 2 (sin A-1
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Virial coefficient
In the case of moderate interactions, the intensity at the origin varies withconcentration according to :
2
I(0)I(0, )
1 2 ...idealc
A Mc
Where A2 is the second viral coefficient which represents pair interactionsand I(0)ideal is to c.A2 is evaluated by performing experiments at various concentrations c. A2 is to the slope of c/I(0,c) vs c.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
In the case of very elongated particles, the radius of gyration of the cross-section can be derived using a similar representation, plotting this time sI(s) vs s2
2 2 2( ) exp( 2 )csI s R s
Finally, in the case of a platelet, a thickness parameter is derived from a plot of s2I(s) vs s2 :
2 2 2 2( ) exp( 4 )ts I s R s
12tR Twith T : thickness
Rods and platelets
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Volume of the particle
21 1(0) ( )i V and
Hence the expression of the volume for a particleof uniform density :
(Porod invariant)
11
(0)
2
iV
Q
21
2
VQ
Hypothesis : the particle has a uniform density
uniform density =>2 2
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
The asymptotic regime : Porod law
Hypothesis : the particule has a sharp interface with the solvent with a uniform electron density
Porod showed that the asymptotic behaviour of the scattering intensity is given by :
1423 48 lim ( ) ( )s s i s S Bs
S is the area of the solute / solvent interfaceB is a correction term accounting for :
- short distance density fluctuations-uncertainties of i(s) at large s (weak signal)
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Distance distribution function
2 2
0
sin(2 )( ) 2 I( )
2
rsp r r s s ds
rs
In theory, the calculation of p(r) from I(s) is simple.Problem : I(s) - is only known over [smin, smax] : truncation
- is affected by experimental errors - might be affected by distorsions due to the beam-size and the bandwidth (neutrons)
Calculation of the Fourier transform of incomplete and noisy data, which is an ill-posed problem.Solution : Indirect Fourier Transform. See lectures by O. Glatter and D. Svergun
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Calculation of p(r)
p(r) is calculated from i(s) using the indirect Fourier Transform methodBasic hypothesis : The particle has a finite size
1 0
sin(2 )( ) 4 ( )
2
MaxD rsi s p r dr
rs
p(r) is parameterized on [0, DMax] by a linear combination of orthogonal functions
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Distance distribution function
The radius of gyration and the intensity at the origin can be derived from p(r)using the following expressions :
and
This alternative estimate of Rg makes use of the whole scattering curve, andis much less sensitive to interactions or to the presence of a small fractionof oligomers.Comparison of both estimates : useful cross-check
max
max
2
2 0
0
( )
2 ( )
D
g D
r p r drR
p r dr
max
0(0) 4 ( )
DI p r dr
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
p(r) :example
Aspartate transcarbamylase from E.coli (ATCase)Heterododecamer (c3) 2(r2)3
quasi D3 symmetryMolecular weight : 306 kDaAllosteric enzyme
2 conformations :T : inactive, compactR : active, expanded
T Rcrys Rsolideal
monodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
ATCase : scattering patterns
10
100
1000
104
0 0.01 0.02 0.03 0.04 0.05
T-state
R-state
I(s)
(a.
u.)
s = 2sin A-1
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
ATCase : p(r) curves
0
5
10
15
20
25
0 50 100 150
T-state
R-state
p(r)
(a.
u.)
r (A)
DMax
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Information content of a SAXS curve
Basic hypothesis : The particle has a finite size Dmax
Sampling theorem : the curve is entirely determined from the values of the intensity at the nodes of the lattice sk = k/2Dmax where k is an integer.
Number of independent structural parameters :
min2 ( )Max MaxJ D s s
Example : spherical protein with 0.3 hydration (w/w) v=0.73 cm3.g-1
Dmax = 1.5 M1/3 => J=3.0 M1/3(sMax – smin)
for smin –=0.002 A-1, sMax =0.05 A-1 and M = 105 => J = 7
idealmonodisperse
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
The case of the sphere
Unique case in which an analytical expression of the scattered intensity can be derived :
2
3
sin cos( ) 3
x x xI s x=2πRs
x
R : radius of the sphere
I(s) = 0 for values of s solutions of 2Rs = tan(2Rs)
first minima at Rs = 0.7158, 1.23, 1.73, etc.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
0
0
0
0
0.01
0.1
1
0 0.5 1 1.5 2 2.5 3 3.5 4
I(s)
sph
ere
Rs
Scattering intensity of the sphere
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
F. Rey, V. Prasad et al., EMBO J. (2001)20, 1485-1497
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
1
10
100
1000
10000
100000
0 0.005 0.01 0.015 0.02
DLP
VLP 2/6
I(s)
s = (2sin Å-1
Rotavirus derivedparticles
DLP : double layer particlescontaining RNAVLP2/6: Virus-like particlesdouble-layer VP6 and VP2About 700 Å diameter
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
1
10
100
1000
10000
100000
0 0.02 0.04 0.06
DLP
VLP 2/6
I(s)
s = (2sin Å-1
Rotavirus derivedparticles
Icosahedral symmetry on thecapsid surface : strong modulation of the electron density => modulation of I(s)
around 25-30 Å :signal associated
with the RNA layers inside the VLP.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Rotavirus DLP
RNA
F. Rey et al., EMBO J. (2001), 20, 1485-1497
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Scattering by an extended chain
In the case of an unfolded protein : models developed for polymers
Gaussian chain : linear association of N monomers of length l with no persistence length (no rigidity due to short range interactions between monomers) and no excluded volume (i.e. no long-range interactions).
Debye formula : where
I(s) depends on a single parameter, Rg .
Valid over a restricted s-range in the case of interacting monomers
Limit at large s :
2
( ) 2( 1 )
(0)xI s
x eI x
22 gx R s
22
2 2
1 1 (2 )lim [ ( )]
2g
sg
R ss I s
R
I(s) varies like s-2 instead of s -4 for a globular particle (Porod law).
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Guinier plot : NCS heat unfolding
Neocarzinostatine : small (113
residue long) all- protein.
arrows : angular range used
for Rg determination
Pérez et al., J. Mol. Biol.(2001)308, 721-743
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
arrows : angular range used
for Rg determination
Debye law : NCS heat unfolding
Pérez et al., J. Mol. Biol.(2001)
308, 721-743
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Kratky plot
This provides a sensitive means of monitoring the degree of compactness of a protein as a function of a given parameter.
This is most conveniently represented using the so-called
Kratky plot of s2I(s) vs s.
Globular particle : bell-shaped curve
Gaussian chain : plateau at large s-values
but beware: a plateau does not imply a Gaussian chain
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Kratky plot : NCS heat unfolding
In spite of the plateau,
not a Gaussian chain when unfolded.
Can be fit by a thick persistent chain
Pérez et al., J. Mol. Biol.(2001), 308, 721-743
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Extended conformation of a modular protein :Hemocyanin from O. vulgaris
O. Vulgaris hemocyanin (Hc), a dioxygen binding protein, is a decameric structure having the shape of a hollow cylinder with a five-fold symmetry axis.
At alkaline pH, or at neutral pH in the presence of a divalent cation chelator (EDTA), the protein dissociates into its constitutive subunits.
Each subunit comprises seven functional units of ca 50kDa joined by flexible linkers.
Study of the dissociation product (subunit) at pH 7.5 and 9.5.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Extended conformation of a modular protein :Hemocyanin from O. vulgaris
1
10
100
1000
0 0.05 0.1 0.15
pH 7.5pH 9.5
I/c (
a.u
.)
q (A-1)
pH 7.5pH 7.5
pH 9.5pH 9.5
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Hemocyanin from O. vulgaris
pH 7.5pH 7.5
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250
p(r
)
r (A)
Rg= 62.5 AMW=400 kDa
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
0 0.0002 0.0004 0.0006
ln(I
/c)
(a.u
.)
q2 (A-2)
Rg=61.7 AMW= 400 kDa
=
Guinier plot Distance distribution function
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Hemocyanin from O. vulgaris
pH 9.5pH 9.5
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
p(r
)r (A)
Rg=102.3 A
6.4
6.5
6.6
6.7
6.8
6.9
7
7.1
5 10-5
1 10-4
1.5 10-4
2 10-4
2.5 10-4
3 10-4
ln(I
/c)
(a.u
.)
q2 (A
-2)
Rg=90.6 A
Slight upward curvature
Guinier plot Distance distribution function
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Hemocyanin from O. vulgaris
pH 9.5pH 9.5
=
0
0.2
0.4
0.6
0.8
1
1.2
0 50 100 150 200 250 300 350
p(r
)
r (A)
Rg=102.3 A
q (A )
Distance distribution function
1
10
100
1000
0 0.05 0.1 0.15
exp.Debye fit
I/c
(a
.u.)
-1
Rg=103.15 A
Debye plot
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Some experimental considerations
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Practical aspects
Small-angle data : Guinier analysis, Rg and I(0)
- requirements : monodisperse and ideal solution to estimate the MW form the I(0) value, the partial
specific volume must be known, and the intensity calibration performed with a reference sample (Lupolen, other protein).
monodispersity : previous check by SEC-HPLC or LS
ideality : extrapolation to infinite dilution by recording experiments at several, precisely determined, concentrations in the range 1mg/ml (or less) – 5 mg/ml according to the MW of the particle.
volume : less than 100µl for each sample (down to 30µl)
- total amount : less than 1mg/ml
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Practical aspects
Wider-angle data : p(r), modeling
- requirements : 5mg/ml <c<10mg/ml with an area detector can provide a reasonable curve with adequate statisitics. If available, a higher concentration will give better data (no stringent monodispersity and ideality requirements)
Note : case of a special sensitivity to X-ray irradiation : - addition of radical scavengers and/or-sample circulation through the beam cross-section (a few µls in the
beam), with a correlative increase in volume.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Lysozyme
Lysozyme : 14 289 Da MW
5 mg/ml solution in Acetate buffer pH 4.5
Rg=15.2 Å in the air
Rg=15.0 Å under vacuum
1000
10000
100000
0 0,01 0,02 0,03 0,04 0,05
Air
[Tam
pon
lyso
l07.
sta/
(3.
07/1
.25)
]*1.
0498
Air
[lyso
l08.
sta
/(3.
07/1
.25)
]*1.
0498
vide
Tam
pon
lyso
s02
.sta
*1.
25
vid
e ly
so s
03.s
ta *
1.25
I(s)
A
s = 2 sin Å-1
1000
10000
0 0,5 1 1,5 2 2,5
I(s)
104 x s2 Å-2
smin
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
PGK
Yeast 3-phosphoglycerate kinase (PGK) : 44 570 Da MW
5 mg/ml solution in Tris buffer pH 7.5
Rg=24.7 Å in the air
Rg=24.4 Å under vacuum
1000
10000
100000
0 0.005 0.01 0.015 0.02 0.025
Z4
2.st
a B
uff
2 A
ir *
4.7
54
6 in
t 25
1-3
63 P
GK
Vid
e *
1.2
5 *
2.8
8
Z4
1.s
ta P
GK
2 A
ir *
4.7
54
6 in
t 25
1-3
63 P
GK
Vid
e *
1.2
5 *
2.8
8
*2
.88
nor
me
lys
vid
e V
ide
Tam
pon
Q1
7.S
HF
*1
.25
(co
rre
c di
am
CA
P)
*2.
88
no
rme
lys
vid
e
Vid
e P
GK
5m
g/m
l Q1
8.S
HF
*1
.25
(co
rre
c di
am
CA
P)
I(s)
B
s = 2 sin Å-1
1000
10000
0 0.4 0.8 1.2 1.6 2
(1/3
)[P
GK
1 S
ub a
ir *5
.448
1 in
t 251
-363
PG
KV
ide*
1.25
*2.8
8 1
/3: p
our
grap
hiqu
e G
uini
erre
glin
(1/3
)[P
GK
1 S
ub a
ir *5
.448
1 in
t 251
-363
PG
KV
ide*
1.25
*2.8
8 1
/3: p
our
grap
hiqu
e G
uini
er *
2.88
nor
me
lys
vide
Vid
e P
GK
- 1
.0 x
Buf
fer
*1.2
5 (c
orr
diam
. CA
P)
reg
lin*2
.88
norm
e ly
s vi
de V
ide
PG
K -
1.0
x B
uffe
r *1
.25
(cor
r di
am. C
AP
)
I(s)
104 x s2 Å-2
smin
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Cell under vacuum
Ratio of buffer scattering
air and windows / under vacuum
5 cm air
25 µm Kapton window upstream
25 µm mica window downstream
0
4
8
12
16
20
0 0.01 0.02 0.03 0.04 0.05
Ra
ppo
rt B
uff
er1
*5
.44
81
/Vid
e*1
.25
ratio
Bu
ff A
ir(c
ol 2
)/V
ide
(co
l9 d
e ly
s-D
ef)
IB
uff
er(A
ir) /
I Buf
fer(
Vac
uum
)
s = 2 sin Å-1
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
S/B ratio
S/B ratio lysozyme 5mg/ml
0
1
2
3
4
5
6
0 0.005 0.01 0.015 0.02 0.025
PG
K1
-Buf
fer/
Bu
ffe
r (s
cale
d!)
Vid
e (
PG
K-1
.0*B
uff
er)
/Bu
ffe
r
I(P
GK
-Bu
ffer
) / I B
uff
ers = 2 sin Å-1
0
1
2
3
0 0.01 0.02 0.03 0.04 0.05
Air
(Lys
-Bu
ff)/
Bu
ffV
ide
(L
ys-B
uff
)/B
uff
I(L
yso-
Bu
ffer
) / I B
uff
er
s = 2 sin Å-1
S/B ratio PGK 5mg/ml
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Limits and drawbacks
SAXS provides spherically averaged intensity, which entails a loss of structural information.
The intensity strongly decreases with the scattering angle, thereby restricting the resolution of the available information . The method is therefore not sensitive to details, to potentially crucial changes in the structure involving e.g. side-chain movements.
The weak signal restrict the characteristic times of kinetics to typically the ms range as compared to the ns or less with Laue TR-PX.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Advantages of the method
SAXS provides time-resolved structural information as well as information on the interactions between particles.
A solution method, SAXS allows one to vary at will most physico-chemical parameters (T, P, pH, ionic strength, substrate, etc.) and assess their effect on the conformation of the particle under investigation.
It can be used as a stand-alone experimental approach (structural parameters, ab initio modeling) or, often most advantageously, coupled to other techniques : structural (EM, PX), spectroscopic (XR or UV-vis absorption, fluorescence, NMR), hydrodynamic, biochemical, molecular biology, etc.
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Background on the PC : 240,200,130But pbs with the HH projector :
Changed to 230,170,120
Background on the PC : 240,200,130But pbs with the HH projector :
Changed to 230,170,120
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Contrast of electron density
solvent
particle
0
el. A-3
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Contrast of electron density
solvent
0 el. A-3
particle
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Contrast of electron densityFormulation of Stuhrmann & Kirste :
P(r) : shape function = 1 inside the particle = 0 outside
The intensity is decomposed in three basic scattering functions
0( ) ( ) ( ) ( ) ( ) ( )P i P i r r r r r
( ) ( )i r r
( ) ( ) ( )P iF F s s s
The shape is observed at infinite contrastOnly fluctuations are observable at vanishing contrastIs and Ii are positive functions which can be directly observedThe cross-term Isi can be positive or negative and cannot be measured directly.
2( ) ( ) ( ) ( )s si iI s I s I s I s
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Contrast of electron density
solvent
particle
0
el. A-3
i
( ) r
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
100
1000
104
105
106
107
0 0.01 0.02 0.03 0.04 0.05
compactswollen
s (Å-1)
Inte
nsi
ty
Kinetics of the Ca2+-dependent swelling
transition of Tomato Bushy Stunt Virus
D24 LURE quadrant gas detector, 1200 s
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Kinetics of the Ca2+-dependent swelling transition of Tomato Bushy Stunt Virus
ID2 ESRF, XRII-CCD detector, 0.1s
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Kinetics of the Ca2+-dependent swelling transition of Tomato Bushy Stunt Virus
Inte
nsi
ty
10.0
1.0
0.10.02 0.04 0.06
Q (A-1)
0.1 s ID2 (ESRF)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Optimal thickness of the sample
This corresponds to a transmission of = e-1=0.37
( )dI de
( )/ 1ddI dd e d
1/optd µ
0
0.1
0.2
0.3
0.4
0.5
8 9 10 11 12 13
d opt (
cm)
Photon energy (keV)
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Optimal thickness of the sample
1/µ
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
I/I op
t
d* = d/dopt
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Optimal thickness of the sample
1/µ
0.6
0.7
0.8
0.9
1
1.1
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
I/I op
t
d* = d/dopt
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EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001
Special case f(r)=g(r)
f( )g ( ) F( )G ( )d d r r r s s s
Parseval’s theorem
2 2f( ) F( )d d r r s s